The theory and applications of fractional differential equations received in recent years considerable interest both in pure mathematics and in applications. There exist several different definitions of fractional differentiation. Whereas in mathematical treatises on fractional differential equations the Riemann-Liouville approach to the notion of the fractional derivative is normally used [3–5], the Caputo fractional derivative often appears in applications [6], Erdèlyi-Kober fractional derivative [7] and The Weyl-Riesz fractional operators [8]. There are some advantages in studying the extremal solution for fractional differential equations, because some boundary conditions are automatically fulfilled and due to lower order differential requirements (see [9]).

Differential equations with maximum arise naturally when solving practical and phenomenon problems, in particular, in those which appear in the study of systems with automatic regulation and automatic control of various technical systems. It often occurs that the law of regulation depends on maximum values of some regulated state parameters over certain time intervals. Many studies of the existence of solutions are imposed such as periodicity, asymptotic stability and oscillatory [10–12]. In [13], the authors discusses the existence of univalent solutions for fractional integral equations with maxima in complex domain, by using technique associated with measures of non-compactness.

In this article, we establish the extreme solutions (maximal and minimal solutions) for fractional differential equation with maxima in sense of Riemann-Liouville fractional operators, by using the Tarski's fixed point theorem. Moreover, we extend the existence of extremal solutions from initial value problems to boundary value problems for infinite quasi-monotone functional systems of fractional differential equations.

2 Preliminaries

The ordered set (poset) X is called a lattice if sup{x1, x2} and inf{x1, x2} exist for all x1, x2∈X. A lattice X is complete when each nonempty subset Y⊂X has the supremum and the infimum in X. In particular, every complete lattice has the maximum and the minimum. Denoted by

[a,b]X={x∈X:a≤x≤b}.

The fundamental tool in our work is the following well-known Tarski's fixed point theorem which can be found in [14]:

Let T > 0 and η > 0 be fixed. We denote by AC([0, T]) the set of all functions x : [0, T] → ℝ which are absolutely continuous and by B([-η, 0]) the set of all functions x : [-η, 0] → ℝ which are bounded. Let M be an arbitrary index set and for each for all ȷ∈M,hȷ:[0,T]→ℝ be a Lebesgue-integrable function and define

Chȷ([0,T])=x:[0,T]→ℝ,|x(s)-x(t)|≤∫sthȷ(η)dη,s,t∈J:=[0,T],

with the property

x1,x2∈Chȷ([0,T]),x1≤x2⇔x1(t)≤x2(t),∀t∈[0,T].

Also, we define the set

Sȷ=ξ:[-η,T]→ℝ:ξ|[-η,0]∈B([-η,0])andξ|[0,T]∈Chȷ([0,T])

satisfies

ξ1,ξ2∈Sȷ,ξ1≤ξ2⇔ξ1(t)≤ξ2(t),t∈[-η,T].

And set

S=∏ȷ∈MSȷ,ȷ∈M

satisfies

γ,λ∈S,γ≤λ⇔γȷ≤λȷ,ȷ∈M.

One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators (see [15]).

Definition 2.1. The fractional (arbitrary) order integral of the function f of order α > 0 is defined by

Definition 3.1. We say that uȷ∈S is a lower solution of problem (1) if for each ȷ∈M we have

Dαuȷ(t)≤Fȷt,ut,maxs∈Jus,t∈J;uȷθ≤ϕθ,θ∈[-η,0].

(2)

Analogously we say that uj is an upper solution of (1) if the above inequalities are reversed. We say that uȷ is a solution of (1) if it is both a lower and an upper solution. A solution u* in A⊂S is a maximal solution in the set A if u* ≥ u for any other solution u∈A. The minimal solution in A is defined analogously by reversing the inequalities; when both a minimal and a maximal solution in A exist, we call them the extremal solutions in A.

Next we pose our main result

Theorem 3.1. Assume that there exist γ, λ∈S with γ ≤ λ such that the following hypotheses hold:

(i)

For each ξ∈ [γ, λ]S the initial value problem

Dαzȷt=Fȷt,zt,maxs∈Jzst∈J;zȷ0=ϕ0

(3)

has a maximal solution z* and a minimal solution z* in A:=[γȷ,λȷ]Chȷ([0,T])

(ii)

For each ξ∈[γ,λ]S,ȷ∈M and t∈J if u(t) ≤ v(t) and uȷ=vȷ then

Fȷt,ut,ξ≤Fȷt,vt,ξ.

(iii)

The function Fȷt,ut,. is nondecreasing in [γ, λ]S. Moreover, the function ϕ is nondecreasing in [-η, 0].

Then problem (1) has a maximal solution, u*, and a minimal one, u*, in [γ, λ]S.

Proof. We shall prove the existence of the maximal solution since the existence of the minimal solution follows from the dual arguments.

Firstly we consider the mapping

Φȷ:[γ,λ]S→[γȷ,λȷ]Sȷ

then in virtue of condition (i) we can define

(Φȷξ)=ϕξ(θ)ifθ∈[-η,0];ξ*(t)ift∈J,

(4)

where ξ* is the the maximal solution in [γȷ,λȷ]Chȷ([0,T]) of the problem (3). Therefore (Φȷξ)∈[γȷ,λȷ]Sȷ. Secondly, we impose the mapping

Φ:[γ,λ]S→[γ,λ]S.

Next we proceed to prove that Φ satisfies the conditions of Theorem 2.1.

Step 1. Φ: [γ, λ]S → [γ, λ]S is nondecreasing.

Let ξ1, ξ2∈ [γ, λ]S and fix ȷ∈M. By (iii) we have

(Φȷξ1)(θ)=ϕξ1(θ)≤ϕξ2(θ)=(Φȷξ2)(θ),θ∈[-η,0].

On the other hand, Φȷξ∈A and in view of conditions (ii) and (iii) we obtain that

(Φȷξ1)≤(Φȷξ2),onJ.

Since ȷ∈M is arbitrary we conclude that (Φξ1) ≤ (Φξ2).

Step 2. [γ, λ]S is a complete lattice.

It suffices to prove that for each ȷ∈M the set [γȷ,λȷ]Sȷ is a complete lattice. Let B⊂[γȷ,λȷ]Sȷ this implies that B ≠ ∅ and B has the supremum and the infimum. Define

ξ*(t)=supξ(t):ξ∈B,t∈[-η,T].

It is clear that ξ*(t) is well defined for all t∈ [-η, T] and satisfies γȷ≤ξ*≤λȷ i.e, ξ* is bounded on [-η, 0]. Finally we shall prove that ξ* ∈A. For fix t, s∈J and ξ∈B we observe that

Therefor ξ*∈[γȷ,λȷ]Sȷ and ξ* = sup B. The existence of inf B is proved by similar manner. Hence [γȷ,λȷ]Sȷ is a complete lattice and consequently [γ,λ]S=∏ȷ∈M[γȷ,λȷ]Sȷ.

Steps 1 and 2 imply that Φ satisfies the conditions of Tarski's fixed point theorem and then Φ has the maximal fixed point x* which satisfies

x*=maxx∈[γ,λ]S:x≤Φx.

(5)

Step 3. X* is the maximal solution of problem (1) in [γ, λ]S.

By the definition of Φ we have u* is a solution for the problem (1). Suppose now that u:=uȷ)ȷ∈M∈[γ,λ]S is a lower solution for (1) i.e.

Dαuȷ(t)≤Fȷt,ut,maxs∈Jusift∈J;uȷ(θ)≤ϕ(θ)ifθ∈[-η,0].

(6)

Then by (5) it follows that for every solution x of the problem (1) satisfies x ≤ x*. This completes the proof of Theorem 3.1.

Remark 3.1. Note that Condition (i) in Theorem 3.1 looks difficult to verify but it is useful for applying the Theorem 2.1. however, there are in the literature a lot of sufficient conditions which imply the existence of extremal solutions. Condition (ii) is called quasimonotonicity. This property is important for extremal fixed points of discontinuous maps. Moreover, the functional boundary condition u(θ) = ϕ(θ), θ∈ [-η, 0] includes the initial condition u(0) = ϕ(0):= u0, where θ = 0. As well as several types of periodic conditions, which have more interest, such as the ordinary periodic condition u(θ) = ϕ(θ):= u(T) for fixed θ which probably takes the value θ = 0. Moreover, the functional periodic condition x(θ) = ϕ(θ):= x(θ + T), θ∈ [-η, 0]. Finally, ϕ(t) can represented as integral initial condition such as

u(0)=∫0Tuȷ(s)ds.

Additional condition on ξ∈S, for all ȷ∈M if ξȷ is Lebesgue-measurable on [-η, 0] leads to suggest the initial condition

u(0)=∫-ηT/2uȷ(s)ds.

Next we replace the condition (i) by assuming F in the set of LX1(J,ℝ×ℝ)-Carathéodory.

Definition 3.2. A mapping p : J ×ℝ → ℝ is said to be Carathéodory if

(C1) t → p (t, u) is measurable for each u∈ℝ,

(C2) u → p (t, u) is continuous a.e. for t∈J.

A Carathéodory function p (t, u) is called L1 (J, ℝ)-Carathéodory if (C3) for each number r > 0 there exists a function hr∈L1(J, ℝ) such that |p(t, u)| ≤ hr (t) a.e t∈J for all u∈ℝ with |u| = r.

A Carathéodory function p (t, u) is called LX1(J,ℝ)- Carthéodory if (C4) there exists a function h∈L1(J, ℝ) such that |p (t, u)| ≤ h (t) a.e t∈J for all u∈ℝ where h is called the bounded function of p.

Theorem 3.2. Let F be LX1(J,ℝ)- Carathéodory. If the assumptions (ii) and (iii) hold then the problem (1) has at least one solution u (t) on J.

Proof. Operating equation (1) by Iα and using the properties of the fractional operators (see [9, 15]), we have

u(t)=ϕ(θ)+∫0t(t-τ)α-1Γ(α)F(τ,u,v)dτ.

Define an operator P as follows:

(Pu)(t):=ϕ(θ)+∫0t(t-τ)α-1Γ(α)F(τ,u,v)dτ.

(7)

Then by the assumption of the theorem and the properties of the fractional calculus we obtain that

where C[(J,ℝ×ℝ)] is the space of all continuous real valued functions on J with a supremum norm ||.||C that is P : Bρ → Bρ . Therefore, P maps Bρ into itself. In fact, P maps the convex closure of P [Bρ] into itself. Since f is bounded on Bρ, thus P [Bρ] is equicontinuous and the Schauder fixed point theorem shows that P has at least one fixed point u∈A such that Pu = u, which is corresponding to solution of the problem (1). To obtain the maximal and minimal solutions, we use the same arguments in Theorem 3.1.

Moreover condition (i) can replaced by letting F in the set of all functions which are μ - Lipschitz. We have the following definition:

Definition 3.3. A function F (t, u, v): J ×ℝ× S → ℝ is called

(i)

a μ - Lipschitz if and only if there exists a positive constant μ such that

F(t,u1,v1)-F(t,u2,v2)≤μ||u1-u2||+||v1-v2||,

where

||.||=supt,s∈J{|.|},

and the constant μ is called a Lipschitz constant.

(ii)

A contraction if and only if it is μ - Lipschitz with μ < 1.

Theorem 3.3. Let F be μ - Lipschitz. If μTαΓ(α+1)<1, then (1) has a unique solution u(t) on J.

Hence by the assumption of the theorem we have that P is a contraction mapping then in view of the Banach fixed point theorem, P has a unique fixed point which is corresponding to the solution of Equation (1). In this case u (t) = u* (t) = u* (t).

It is clear that F is LX1(J,ℝ)- Carathéodory with any decreasing growth function h∈L1(J, ℝ+) such that ||F (t, u)|| ≤ h (t) a.e t∈J for all u∈ℝ. Therefore in view of Theorem 3.2, the problem (8) has maximal and minimal solutions.

Example 3.2. Let S be any nonmeasurable set such that S⊂ [0, 1]. Consider the problem

Dαu(t)=1,ifu>t,t∈J;1,ifu=t,t∈S0,otherwise.u(0)=0.

(9)

Obviously F does not satisfy the condition (i) of Theorem 3.1, and hence the problem (9) hasn't extremal solutions.

Declarations

Acknowledgements

This research has been funded by the University Malaya, under the Grant No. RG208-11AFR.

Competing interests

The authors declare that they have no competing interests.

Authors’ Affiliations

(1)

Institute of Mathematical Sciences, University Malaya

References

Lewandowski R, Chorazyczewski B: Identification of the parameters of the KelvinVoigt and the Maxwell fractional models, used to modeling of viscoelastic dampers. Comput Struct 2010, 88: 1–17. 10.1016/j.compstruc.2009.09.001View ArticleGoogle Scholar

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